Gem Combination
Foreword: In order to better understand the mechanics of gem combination, I made several kinds of testing, with the help of a editor of flash save data. This allowed me to access the raw data used by the game for calculations, set up "experimental gems" specific to the tests, and also look for limit behaviour in higher tiers of combination. Please note that any general rules that I have found and explicited below may still be in fact slightly more complicated; and you might still encounter little irregularities, occasionally. (I tried my best, but it seems they really wanted to make it complicated at parts...) Gem Values The first thing to know about the intricacies of combination, is that the apparent value of a gem is not its "real" value. It can be acknowledged when trying to do combinations with a bunch of gems which are apparently identical (same tier, same stats, same stat values), and observing slightly different results in stat values, depending on the exact gems used. In this article, I will be using a couple abbreviations on the matter of gem values: *"DV": Displayed Value; the value (for one given stat on the gem) which appears in-game when mousing over a gem, and which is applied additively to a minion's stat upon equipping the gem. *"IV": Internal Value; the stat value that the gem really holds, and that the game actually considers for calculations. Relevant Facts *A gem has an IV of at least 1 in any stat it appears to hold. *(Experimentally, forcing a gem's IV to a value lower than 1 (ex: 0.5) results in a "null gem", as I call it: it appears grey in the gem inventory, displays no stat, and when moved around it causes bugs among the inventory slots). *Gems you acquire have their IV generated randomly (within the range of their tier) and this IV is rarely an integer. However, you wont' be able to see it, because the DV you see when checking the gem's stats is an integer... The numerical rounding involved is one of the reasons for irregular behaviours, like the one mentioned above. Conversion Mechanism The actual combination mechanism is actually simplistic: when you put your three gems into combination (be they standard or mixed), all the game does is add the respective stat IV together, into the resulting gem. I.E. The output gem's IV is always equal to the sum of the three input gem's IV (for each stat separately). Example: Let's consider combining the following gems together: *the first one has 100 IV in Health *the second one also has 100 IV in Health, plus 100 IV in Energy *the third one has 100 IV in Health, 100 IV in Energy and 100 IV in Speed Then, the resulting mixed gem would have 300 IV in Health, 200 IV in Energy and 100 IV in Speed. Simple. However, these are only the IV of the gems, and we are more interested in the corresponding DV, the values that are visible in-game and relevant to improving our minions' stats. Short Analysis Well, first things first: At tier 1(T1), a gem's DV is essentially rounded down from its IV. (Ex: a gem with IV=1.2 and another one with IV=1.7 will both display DV=1) But, this does not allow us to simply predict the value of a tier-2(T2) gem combined from tier-1(T1) gems: indeed, combining three +5 (T1) gems does not return a +15 (T2) gem, but a +9 (T2) gem... From this, we understand that there is some sort of "conversion factor" between the IV that we combine and the DV that we get, which is reducing the expected value of the gem. (At least, that is the hypothesis on which I based my testings...) So I tried to gather data of several combinations (DV and corresponding IV, via save editor), and calculate a general conversion factor (that I call "C") that would fit the equation 'IV x C = DV' for these various combinations ; but I could not find a satisfactory one, that would not produce large error margins in DV when reapplied to IV (recalculating the normal way). Thus, I concluded that there was another variable affecting the conversion. And here is the twist: The conversion factor C actually varies depending on the tier (for a tier "N", it becomes a different "C(N)"). Specifically, C decreases for higher tiers (starting from tier 1). But that's not all ! There is another quirk: it also appeared that C slightly decreases for higher IV within a given tier (but it is a much smaller decrease than between tiers ; this could perhaps be due to heavier rounding for lower values...) We can keep in mind that, at tier N: DV = IV x C(N) Experimentation Tier Tables At first, I gathered data by "farming" gems from the shop and noting the result for a bunch of specific combinations. As per irregular behaviour mentioned below, there were systematic inconsistencies: combining different gems with the same sum of DV generally returned one specific value, but once in a while returned a slightly different value (different, by 1 stat unit usually). This is most likely due to the rounding applied to the IV during conversion, and is problematic when starting from the rounded DV to calculate the corresponding IV... Note: I made the testing with 5-10 samples of each gem value. So, I may not have encountered all possible alternate values for each DV sum... I checked triplets of standard gems with identical values, withing tier ranges (from the shop). For the first few tiers, I checked all the values ; for later tiers, I only considered some arbitrary values: the minimum available, multiples of +5, and the maximum available. (As the output result is proportional to the input sum, it is possible to extrapolate the intermediate values in a linear fashion, with good enough precision.) I compiled these results here as quick reference data. *On the left are the input values (DV). Within a tier, we first consider the case of putting only one gem (Gem1) of the considered stat into combination, the other two being assumed to be of other stats ; the result is the value of the component of the first gem's stat in the mixed gem. Then, the case where the first two gems (Gem1,Gem2) are of the same stat, and the third one is of another. And finally, the case where all three (Gem1,Gem2,Gem3) are of the same stat, giving out another standard gem. *In the middle is the sum of input stats (DV), for comparison and extrapolation. *On the right are the found output results. As mentioned above, several different outputs may be obtained; note that here, they are simply recorded in increasing order ; not by probability of appearance (i.e. Result1 is not necessarily the "main" value you will obtain, nor is Result2 a "rarer" value). Tier Factors Afterward, in an attempt to find out the respective factors for every tiers, I made some testing over various samples of IV. When I say "various", I mean that I edited in some "milestone" values at different scales of numbers (1,2,3... then 10,20,30... then 100,200,300... then 1000,2000,3000... etc). I did so because I know the IV values are growing very large between tiers (for 3 identical gems combined, it's like the IV is multiplied by 3 each tier) ; obviously, this is not the same for DV, which are growing at a much more "reasonable" pace (due to factor C decreasing itself between tiers). Here are my attempts at approximating the conversion factors for each tier. I also included a "simplified" factor, for ease of use, which is essentially a rounded version of the main factor, and at the same time the version that should be most accurate within the range of values specific to the tier (considering the factor's slight decrease at greater values). Remark: By comparing the evolution of the factors between each tier, it looks as if each one were divided by a number (~2 at first) which increases after each tier: *That would be 1.8 at tier 2 (from tier 1); *~1.9 at tier 3 (from tier 2); *2 at tier 4 (from tier 3); *~2.1 at tier 5; *~2.2 at tier 6; *~2.3 at tier 7; *~2.35 at tier 8; *~2.4 at tier 9; *~2.45 at tier 10; *(or something like that). The increase in this divider appears to slow down as we progress through tiers. And we can extrapolate that "something" will happen if the divider becomes greater than 3, because......... ...Well, I am not sure how to correctly demonstrate it mathematically, but here is the intuitive idea: Once again, let's suppose we are combining only identical standard gems (with the same type and stat), which we reuse at each tier for the next combinations (would take lots of gems of course). Then, it's like the input value is tripled after each combination/tier; but the output result is "moderated" by the divider (we multiply by 3, then divide by ~2). For the main tiers (those we have normal access to), it is beneficial in terms of stat gains, since 3/2=1.5 >1 i.e. we multiply the effective result by 1.5 and thus gain a number greater than the one we started with. However, if the divider were to reach 3, it means that we would triple the input then take the third as result, and gain no stat benefit. Furthermore, if the divider exceeded 3, we would end up with a losing ratio (<1) at this tier and the following i.e. the stat value will start to actually decrease through combinations, until at some (very high) tier, the combination returns a gem of +1 (the minimum). So, according to this reasoning, the tier where we get the last divider still inferior to 3, should indicate us that this tier is the highest beneficial tier of gems, where we will be able to imagine the strongest gem possible in-game. Note: Keep in mind that the divider I am considering here and the previous factor C both describe equivalently the same mechanics. The difference is that the factor C is a set multiplier applied to IV, specific to each tier, while the divider is more of a comparative multiplier between tiers (i.e. the derivative of C). To find out if the above conjecture is exact, have a look at the next section: "Limit Behaviour".. If you are not so interested, and rather want a little insight on how to handle all the previous data, head on to the last section instead: "Tips".. Limit Behaviour The Limit So, for the sake of curiosity, I experimented editing gems of high tiers, to find out a "maximum (beneficial) tier", as per the previous conjecture (and also to have a look at the behaviour of even higher tiers). And the results are: *There is indeed one such "maximum tier", but it's not actually situated where the derivative of C approaches 0.333 (rather, around 0.375...). * In addition, the derivative does not simply keep on decreasing after this tier; instead, the factor C starts having rather impredictable behaviours, resulting in generally less-than-useful gems (see next subsection). All in all, the max/last "reliable" tier is Tier 17. According to calculation and testing, using standard T1 gems of +5 (the apparent maximum DV available from the shop at the beginning of the game), makes it possible to obtain higher maximum-value gems than the best (rarely) available for the shop. For example, one could obtain up to +79 (T9) gems, which would combine for +95 (T10) gems ; while the best T9 gems from the shop (+69) cannot produce more than a +83 (T10) gem... So, by combining (huge amounts of) such identical +5 gems, one could theorically achieve a maximum standard T17 gem of +259, perhaps even more (~260) if the T1 gems actually have a rounded DV slightly superior to 5 (5.1, 5.2,...). As a comparison, I calculated the maximum for a standard T17 gem that would be made from combining a bunch of T9 gems acquired from the shop ; however, I considered gems of +66, which is a bit more reasonable than taking the +69 ones which are the best available (and would take ages to farm from the shop) ; and the result is +217 (more if you include some gems higher than +66). If you are interested in mixed gems, I also pulled out a few tests: with three types of +5 (T1) gems, you could (theorically) achieve a three-stats T17 gem of up to +87/+87/+87 ; with +66 (T9) gems, you could get a +73/+73+/73 one. Here is a small table summing up my findings about Tier 17. (Minima of tier range where calculated considering +2 (T1) gems, and maxima with +5 (T1) gems.) Remark: I started this section by saying "for curiosity", because obviously, the quantity of gems needed to craft a Tier 17 gem would require an insane amout of farming time (let alone the culling of lower-value gems): *If you started at tier 9, it would take you 6561 gems (and $ 16,632,135) to reach Tier 17. *And if you considered starting from tier 1 (to get the best of the best), that would be 43,046,721 gems '''(and $ 258,280,326)... Over-Limit As mentioned previously, tiers above the threshold of 17 start becoming unreliable in terms of combination... Notably, tier 18 has quite an erratic behavior: *For low IV values (relatively to tier range), the combination/conversion appears to work normally: resulting DV seem consistent compared to T17 values. So far, so good. *For most intermediate values however, things start to get messed up: the input IV is converted into (massive) negative values ! It then returns what I call a "negative gem": although it looks (and moves around the inventory) normally, it shows no DV and has negative stat bonus', '''reducing your minion's stat ! These stat malus' start out quite strong (around -120), increasing back towards 0 following increase in input IV. (Example: By editing in a T18 gem made of low-value +33 (T9) gems, I obtained a "negative gem" of -117.5.) *For the highest values (within range), as a continuation of the previous behaviour: the input IV is harshly reduced (divided by some significant numbers, like 3 or 9, depending), and the DV is affected equally: you end up with T18 gems of +40~50 or so... So overall, tier 18 is not really reliable for combination (even theoretically). Remark: There might still be a particularity that could render tier 18 interesting after all: mixed T18 gems (for instance, 3-type ones) may have stat values that are individually lower that the threshold for negative output (around +116 DV), allowing them to work normally for values not achieved by T17 mixed gems. Indeed, it seems you could (theorically) achieve a maximum 3-mixed T18 gem of +97/+97/+97 (which could in fact be more "stat-efficient" than the standard T17 gem of +259). Highest Non-null Tiers Tier 19 sports roughly the same oddities as tier 18, but it starts outright in the "high-value crippled" behaviour, which is exacerbated (IV are sometimes divided by even larger numbers, like 122) ; also, with a quirk: the resulting low DV (~40) are (randomly?) switched with their negative counterpart... At tiers 20 and higher, the behaviour becomes somewhat clearer: any input IV defaults to one large negative value (-268435456), independent from tier. This returns "negative gems"(no DV), with a constant stat "bonus" (or rather "malus") at each tier, (for any IV input, standard or mixed gem) ; however, the resulting malus' still differ between tiers: *At tier 20, the "malus" is around -16, but may be of opposite sign in certain cases. (Actually, in these cases, the IV is set to approximately the opposite of the default (huge) value, thus returning a rather low positive value: +17.) *At tier 21, you get a regular -5 malus ; and at tier 22, it's -2. (From there on, IV is always forced into the default negative value.) *Tier 23, this time, returns a "negative gem" of +0, no matter what (at last). (I have even tried a few higher tiers to be sure, and definitely, this is the final limit.) Note: As I said, the above malus' are stable ; however, one might be misled into thinking they are not, due to this small anomalie (bug?), where every stat bonus applied to Energy turns out to be worth 1.5x its DV... Negative Stats A note on the usability of negative stat bonus': Equipped with "negative gems", a minion may actually enter battle with negative stats. Depending on the stat affected, one of the following may ensue: *Negative Health: the minion does not appear on the battlefield, as it is considered defeated from the beginning of the battle.(Way to lose a battle before even starting it) *Negative Energy: the minion cannot act at all, not even use desperation.(Way to stuck oneself in battle) *Negative Attack: the minion can still deals a (small?) amount of damage.(Perhaps due to a "default base damage" in skills) *Negative Healing: the minion applies negative healing to its target, meaning that it will actually damage its target by healing them (including itself). *Negative Speed: intuitively, the minion will always act last in turn, as it has the lowest speed value on the battlefield. Remark: One may consider the opportunity of 'negative Speed': if one has built their strategy in such ways that one (or several) of their minions are required to act the latest possible in turn, then possibility of reducing their Speed stat would be tactically beneficial. (But again, crafting gems of tier 18 or higher would take an unreasonable amount of time, stalking the shop and replaying trainers to death for their gold...) Tips To conclude this article, I will give a little practical insight about using the previous data (although a reader coming to this page would probably have an already precise idea of what information they are looking for, and what for). Example of Usage Firstly, the most "usual" subjects of gem calculation I can think of, would be either: '' 'I wonder how much stat bonus I would get from combining these specific gems that I already possess.' '' , or: '' 'I plan to combine some tier "T" gems, in order to obtain a tier "T+1" gem with a specific wanted DV in a given stat ; and I want to know what gems of tier T will be required.' '' From these premises, you can follow two procedures to get an answer: 1°) Simply look up the Tier Tables: *See what values a certain combination (in the "Sum" columns) would give you. *Or, search for the value you want to achieve in the "Result" columns to get the values you would need. *(If the sum or needed value provided in the tables are not spot-on, you can approximate the desired data from values before and after in the table ; a linear approximation should suffice.) 2°) Calculate the desired data according to the conversion mechanics: *Retro-convert the DV of your tier T gems into an IV sum (applying 1/C(T) to DV) ; then convert this IV sum into a tier T+1 DV (applying C(T+1) to IV) to predict combination results. *Or, knowing the DV you want to achieve and its possible tier T+1, retro-convert this DV (with 1/C(T+1)) into a tier T wanted IV sum ; then consider for instance the third of this sum as the average IV you will need in your three tier T gems, and convert it (with C(T)) to the average DV you will need. *(Although looking up the Tier Tables is simpler, this method may be more accurate for values not directly porvided in the tables ; and it can be used for extrapolations.) Note: Remember that, due to rounding of the real gem values, there always can be a small uncertainly in the combination result. In general, it doesn't amount to more than 1-3 units (although it may be amplified if applied to the stat for which your minion has a +5% affinity, bringing the uncertainty to 2-6 units). Example: Let's say I want to craft a mixed gem of +50 Healing and +20 Energy. By looking up the Tier Tables: *I find that a +20 bonus can be achieved at T10 with one standard T9 gem of roughly +50 (or slighly less); *and that +50 at T10 will require two standard T9 gems between +60 and +65. For the +50 part, I can try to work with some T9 gems in-between the two values, or verify by calculation: *I start by retro-converting the two wanted DV into wanted IV, at tier 10 where C(10) ~= 0.000966; *on the one hand: IV(20) = 20/C(10) ~= 20704 (needed IV from one T9 gem); *on the other hand: IV(50) = 50/C(10) ~= 51760 (needed IV sum from two T9 gems). *Then I convert these needed IV into needed DV, at tier 9 where C(9) ~= 0.0023888 ; *DV(20) ~= C(9)x20704 ~= 49.5 (in one T9 gem); *DV(50) ~= C(9)x51760 ~= 123.6 (over two T9 gems i.e. 61.8 average per gem). Thus, I will need to shop for two T9 gems of +62/+61 Healing (on average), and one T9 gem of +50/49 Energy. Obtaining a +100 Gem Lastly, I figured I would test what gem values and tier are needed to combine for a standard gem of +100 (as a kind of reference value). As stated previously, the maximum we can get at tier 10 is +83 with T9 gems from the shop, or +95 with +5 (T1) gems. So intuitively, we understand that we are close ; we only need to climb a few more tier to reach the goal... (Note: I reasoned on conversion calculation, by extrapolating conversion factors for the following tiers, in order to orient my testing values.) ...And indeed, the testings reveal that a standard gem of +100 can theorically be achieved at tier 11 ...but, using +5 gems from tier 1. It is just not possible with T9 gems from the shop: +100 (T11) requires at least +84 (T10) ; versus the maximum of +83 (T10), which seems to cap at +99 (T11) after combination... Alternatively, by pushing yet a little further, such a +100 gem is achievable at tier 12, by using T9 gems of +59 and higher. These are rather high in the tier range, but not too much ; so hopefully they should not show up too scarcely, which would make for a reasonable farming (at least, more reasonable than trying to get only the best in tier). As for the expense, that would cost $ 68445 for the 27 gems needed. Category:Gameplay